So in that case (of a single zero-measure box, all of whose sub-boxes lying in the sub-copula have measure zero), you can do it.
One way to do it is to take a copula extending the sub-copula and then adjust. The way that I would do this is consider each atomic box $D$ from the sub-copula that intersects $\partial B$ [ by atomic, I mean a box of the form $[a,b]\times [c,d]$ with $a,b$ consecutive elements of $\mathcal S_1$ and $c,d$ consecutive elements of $\mathcal S_2$ ]. I would then uniformly the distribute mass from $D$ to $D\setminus B$. This gives a new copula which agrees with the previous copula on all of the atomic boxes, hence is an extension of the copula.
This can also be made to work by induction if there are a number of boxes to be assigned zero measure with gaps between. But it's not hard to see it can fail if one of the atomic boxes lies in the union of the zero-measure boxes, but not in any single zero-measure box.
More generally, I think you need a version of Hall's "Marriage Lemma". This is sometimes stated in terms of "men" marrying "women" (or points on the $x$-axis being coupled with points on the $y$-axis). I will use that language here, as it is easy to read, even though it suffers from lacking societal awareness. If there are the same number of men and women (corresponding to there being a probability distribution on the $x$- and $y$-axes), then there is a matching (corresponding to a copula) if and only if each subset $A$ of the men is collectively friendly with (i.e. the pair lies outside the zero-measure boxes) at least $|A|$ women; and each subset $B$ of the women is collectively friendly with at least $|B|$ men.
I would have to think in a bit more detail to see if I could formulate the precise condition analogous to the marriage condition in terms of copulae. Also, the Hall Marriage Lemma deals with two axes. One way to think of this in terms of the Hall Marriage Lemma is in terms of bipartite graphs: there are edges between a man and a woman if they are compatible. One is looking for a bijection. There are hypergraph versions of the Hall Marriage Lemma, which might allow one to deal with $n$-dimensional copulae, but for the moment, I have no idea how this would work.